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In March, I gave a talk to some math textbook authors describing five strategies for designing curricula for digital media like tablets and computers. One of those five strategies relatedly directly to the ladder of abstraction and my tentative hypothesis that paper is a problem, that the constraints and cost of paper lead us to decisions that ultimately make the process of abstraction very difficult for students to understand. (ie. Print-based curricula in your teens leads you to tell people that “Math always seemed abstract to me” in your thirties.)

My preference would be that you’d watch the 8.5 minutes from 19:14 to 27:57 of this video. Some of the examples don’t work well here in text, but I’m going to lay out the slides and narration anyway so we have a place to argue about this segment in specific.

This gets really fun here. This is a part I’m really enthusiastic about right now. Have students climb the entire ladder of abstraction.

This is NCTM on technology and I think they got it exactly half right. The technology helps us work at higher levels of abstraction but technology also helps us work at lower levels of abstraction. Right now, the tasks we give students are focused on this narrow band, this narrow set of rungs in the middle of that ladder and no higher and no lower.

You look at the modeling standard and look at what it says. (This terrifies me by the way.) “Identifying variables. Formulating a model. Analyzing and performing operations. Interpreting the results. Validating the conclusions.” That’s your ladder of abstraction there. [I won’t exactly sign on off that now, FWIW, but let’s see where he’s going with this. –dm] And what do we have students doing? Just that middle rung.

They select the operation and apply it. That’s it.

And so this is a tool that was built by a guy named Bret Victor. This is a guy you should get to know. I watched a talk he gave on the plane over here from San Francisco and I had like a physical reaction to this talk it was that good. He has a project called Kill Math, which should be provocative enough for all of you guys to click on the link. He’s a technologist, a creator, an artist, an engineer. This is a guy who will provoke your thinking in a number of different ways.

But he created this tool he calls Tangle and I adapted it for use in this problem. And what it lets me do is essentially turn all those parameters into variables and so now once I’ve solved the first problem, I can slide around on all those things and see what would happen if gas dropped in price or if the shuttle costs went up or if the cost of parking in Santa Rosa dropped to zero. And ask a whole host of new, different, complicated questions. And pose scenarios that are higher up on the ladder of abstraction.

But we also need to have students work at lower levels. Like, where did those parameters come from? You and me. We brainstormed them [earlier in the session –dm]. But in the textbook those are given to them. The text tells students what parameters they’re going to need and it gives them that information.

Thats a valuable lower rung on the ladder that students need experience with. Once we have the task posed, let’s just ask that student, “What information will you need here?” And just let the student think about it for a second, and then type a few things down in a low-risk environment.

And maybe let the student see all the classmates’ responses also, the results of that brainstorming.

That’s a lower level of abstraction. You and I are constantly dealing with the question, “What information do I need to solve this task?” That’s a question that gets very little air time in our print-based curricula.

Again, we have a third page for this problem now. We have the starter — the context, the visual. We have this rung — this level of abstraction. And then the rest of the problem. That’s three pages for this task right now. You can’t do it in print.

We could compress all of those. But you can’t put a kid on a higher rung and then ask them to work on a lower one. Like I can’t give the kid all the information they’ll need and then ask them “what information will you need?” The horse is out of the barn. So we have to split this up over multiple “pages”.

Go even lower. This is, I think, the lowest rung on the ladder of abstraction.

We show. We don’t tell. And then we pose the task immediately, “Does the ball go in?” And, at this point, how are you not speculating? Like how are you not guessing. I know you have a theory. You’ve got an idea in your head whether it’s going in or not. Can I get you guys to raise a hand if you think it’s going in? Okay. And the rest think it’s going out. I’ll just assume that. That is a valuable moment of intuitive and guesswork and it’s engaging for students so let’s give them an outlet for that on the page there.

Just let them tap a yes or a no. “What do you think?” And then we’ll aggregate all those responses.

This has cost us nothing in terms of cash or weight on the student’s back. It’s cost us a brief second of class time, which makes whatever meager return we get on this investment just incredible. Just in terms of, “Well now I want to know the answer.” We’ve got that student.

And that’s everywhere, in applied math particularly. “How many minutes will it take to fill?” Every student puts down a guess before we get in the real meat of it.

This is going to get really fun here. Obviously I will ask you, “How deep do you think it is?” That will happen. But even better is this. I’m going to ask you to tap on the screen when you think the rock hits the ground. And I can’t have you do that. [This part makes no sense on paper but it’s pretty awesome in the video –dm] But I’m going to ask you to raise your hand when you think the rock hits the ground.

Now tapping obviously has a lot of advantages over raising your hand. It’s more surreptitious. You have your own answer. You’re less biased by others. I like that about it. But for now you saw that video before. I’m going to ask you to raise your hand up when you think the rock hits the ground.

[Most people raise their hand on the huge boom. I raise my hand way earlier than the boom. –dm]

We were all over the place there. There was some large clumps there at the end. I think the you guys were late. I don’t know what the answer is, exactly, but I know that this crowd here, on that big boom, was late. Why? What was the question that these people answered perfectly? “When did I hear it?” Which is different than “When did it hit the ground?” Right. Because the sound is coming back up.

And you guys should see your faces right now. Some of you guys are kind of like, “Huhhh.” So using a very low rung — which student couldn’t answer that question? — we’ve highlighted that there’s more here than projectile motion. There’s also the speed of sound going on. We’ve got that in your head.

And thinking about how that happens in print, we get a very different reaction off this right here when we’re just writing and writing and writing about the speed of sound. It’s different.

This plays out in pure math in some fun ways. This is your very traditional trigonometry practice unit. We jump into problems like this right here.

Calculate the missing side. And we do this very interesting thing on these problems where the student gets an answer and we go around and we look at the answers. When the answer is wrong, we ask, “Does your answer make sense? Is your answer reasonable?” It has the effect of stigmatizing that question, of course, so they know “Is your answer reasonable?” means “Your answer is wrong.” But it’s interesting when we ask that question. We ask that question once they’ve climbed to the very highest rung on the ladder. They’ve selected a model. They’ve performed the operation. They are way high up there. And we ask them to climb on down and access the lowest rung on the ladder. “Is your answer guessable, reasonable? Does your intuition say, ‘Yeah, it works.'”

I don’t have a ton of evidence but I think that’s hard for students. Far better would be to start the student at that low rung and then build up. So I’m talking about this right here:

We just ask them to estimate it at the very start. All the same problems. They’re just going to see them twice, though. They’re going to go through the first time in a couple of minutes and just put down their best guess at how long that side will be. So maybe the student says, “55, maybe,” and moves onto the next one. It takes two minutes.

Then the next time through, we give them this angle here. Their answer — their intuitive answer — is still up there so that when the student gets that it’ll be 49.7. Like, yeah, that’s kind of within the ballpark there. And we don’t have to answer the question, “Is that reasonable?” Their reason is staring them in the face already.

So if they were to use the wrong identity. It would be more obvious. So if the student got 14, or whatever it would be, if they had solved accidentally for the other side, it might be more obvious seeing that guess staring them in the face.

10 Comments

I’m excited about this because it is easily accessible for me as a teacher. This is something I could do tomorrow, if I was teaching trig tomorrow. Even without students having access to the technology, since I have a projector, I can make the numbers appear when I want to through PowerPoint.

It does make perfect sense to ask them to guess first. It’s even a pretty decent hook with minimal work and could be a smooth transition into the lesson.

I’m excited to get started. What great timing to post this simple idea at the beginning of the school year.

I love the question you asked when people put their hands up indicating when they think the rock hit the ground: “what question are you answering perfectly?” I thought hard about why the question “when do you hear the rock hit?” question was different from the “when does it hit the ground?” question in a way that I wouldn’t have if you had just said, “You’re wrong, that’s when you hear the rock hit the ground. I asked when does it hit the ground.”

And I think it’s also worth noticing that you just told the people who put their hands up early, “oh, that’s the rock hitting the side of the well.” Sometimes it’s okay to just tell people stuff about the context.

William

It’s always surprised me that a third grade math textbook is structurally identical to a twelfth grade math textbook. In other subjects, we recognize that the material students play with changes qualitatively as they get older.

Imagine if we never exposed our students literature, to Shakespeare or to Milton or to Vergil or to Homer. They’re part of the reason that you learn the formalism and the operations of the languages that you’re learning.

What you’re suggesting, I think, is that way we write curriculum is equivalent to the pablum that ends up as examples in English (or foreign language!) textbooks. It’s stripped of the immediacy, tension, drama, and truth and resonance of good literature. We can do better.

I stayed up till 2:00 in the morning to watch Curiosity land because it was the answer to a heck of a math problem. Even though I didn’t do the work to solve it, the drama was killing me. I *had* to know the answer.

Dan, I love what you’re doing here, but I wonder if it doesn’t coincide with what I’m working on, which is essentially the same thing you are describing, but with more iterations and more tools (especially at the start).

So instead of guessing about the basketball once, you have many many opportunities to guess (different shots). Not just multiple guesses as more information is added, but multiple problems with slight variation (not sure if you already intended that).

I’m also encouraging more tools so that you can literally get the right answer through bulk force. So for the basketball problem maybe there’s a way to create an additional arc through a simulation. You can see if that arc matches the current trajectory and goes in. As you progress through more problems the tools get more complex, so maybe you have to type in the calculation required to provide the estimated arc and see if it matches up.

Debbie

Hi Dan, this post makes me wonder a couple of things. Firstly, are any publishers actually listening to you? Is there a project under way to produce a textbook along these lines?

Secondly, we keep seeing the same examples – the water butt, the basketball shot et cetera. Great questions which I feel exemplify your thoughts so clearly, but do you find that you are spending less time on this element of classroom teaching?

@Jared, I’d be interested in hearing more about your vision for PuzzleSchool. When I first checked it out, it looked like you were aggregating other puzzle-type games but this sounds more ambitious here.

Debbie:

Hi Dan, this post makes me wonder a couple of things. Firstly, are any publishers actually listening to you? Is there a project under way to produce a textbook along these lines?

I’ve been advising Pearson on their new CCSS textbook but I’m afraid, as a sub-contractor of a sub-contractor, I’m pretty easy to ignore.

Secondly, we keep seeing the same examples – the water butt, the basketball shot et cetera. Great questions which I feel exemplify your thoughts so clearly, but do you find that you are spending less time on this element of classroom teaching?

There’s never enough time for everything, unfortunately. The more I give talks about curriculum design, the less time I have for designing curriculum. I release a couple of new tasks every month at threeacts.mrmeyer.com, though.

Debbie

I think we underestimate the power of the guess, tap, estimate in the math classroom. I was struck by your trig example, what is the point of asking a student if their answer is reasonable at the end of the end of the problem if we never asked them to consider a reasonable length to begin with?

I had my first day with students today and we used your Gatorade problem with our Algebra I classes for the second year in a row. I was surprised when more than half the class thought the 6-pack was going to be the better deal, but it was that much more powerful when they proved themselves wrong with the math!

The new technology that was added from last year’s Gatorade problem to this year’s was the introduction of the document camera. It is amazing how such a simple tool can change the conversation about math in your classroom when 23 ninth graders are all looking at and analyzing their classmates’ work.

Have any of you seen the video/film MAA did of Polya teaching a group of students at Stanford in the ’60s – “Let Us Teach Guessing”? He gives them a very nice problem about 5 planes subdividing 3-space into regions and the way things progress during the lesson is masterful. His accent is sometimes hard to parse, but it’s very much worth the effort.

Someone recently put it up, but I’m waiting for permission to share the link. I hope everyone interested in the issues being pursued here gets a chance to see it.

That said, I have a specific question for Dan: you say

“You look at the modeling standard and look at what it says. (This terrifies me by the way.) “Identifying variables. Formulating a model. Analyzing and performing operations. Interpreting the results. Validating the conclusions.” That’s your ladder of abstraction there. [I won’t exactly sign on off that now, FWIW, but let’s see where he’s going with this. -dm] And what do we have students doing? Just that middle rung.”

Could you elaborate on what terrifies you exactly in what you’re citing and why? Not that I don’t think there are things to be terrified by, but I’m not clear as to what you’re pointing there and don’t want to jump to conclusions about your reasons, either. Thanks.